Engraftment of viable, electrically functional cells into a myocardial infarct as a method for restoring functionality is currently a topic of active research interest. Cells implanted in this way can form gap junction connectivity with each other, but often do not connect well with the surrounding tissue outside the infarct. Using a bidomain computer simulation model, we find that activation of these implanted cells by outside propagating action potentials is nevertheless possible, even if no gap junction connectivity to the surrounding tissue exists at all. The mechanism by which this action potential "tunneling" process occurs involves a current path that passes through both the intracellular and extracellular spaces, and is fundamentally spatially two-dimensional in nature. The typically convex boundary of the region occupied by these cells is found to greatly enhance the tunneling process, but unfortunately also hinders the ability of the activation of these cells to terminate reentrant waves propagating around the infarct.